12.2 Calculus of Variations 675
Hence the shape of the solid body having minimum drag is given by the equation
y(x)=R
(x
L
) 3 / 4
12.2.3 Lagrange Multipliers and Constraints
If the variablexis not completely independent but has to satisfy some condition(s) of
constraint, the problem can be stated as follows:
Find the functiony(x)such that the integral
A=
∫x 2
x 1
F
(
x, y,
dy
dx
)
dx→minimum
subject to the constraint (12.17)
g
(
x, y,
dy
dx
)
= 0
wheregmay be an integral function. The stationary value of a constrained calculus of
variations problem can be found by the use of Lagrange multipliers. To illustrate the
method, let us consider a problem known as isoperimetric problem given below.
Example 12.3 Optimum Design of a Cooling Fin Cooling fins are used on radiators
to increase the rate of heat transfer from a hot surface (wall) to the surrounding
fluid. Often, we will be interested in finding the optimum tapering of a fin (of
rectangular cross section) of specified total mass which transfers the maximum
heat energy.
The configuration of the fin is shown in Fig. 12.5. IfT 0 andT∞denote the wall
and the ambient temperatures, respectively, the temperature of the fin at any point,
T (x), can be nondimensionalized as
t (x)=
T (x)−T∞
T 0 −T∞
(E 1 )
so thatt ( 0 )=1 andt (∞) =0.
Figure 12.5 Geometry of a cooling fin.
